The following theorem, which relies both on numerical properties of \(\Phi _{ef}^{\ast }(p)\)  and on Lemmas REF  and REF ,
states that either \(X_{x}\)  is a large subgroup of \(X\)  or \(\Phi _{ef}^{\ast }(p)\)  is a square, or \(\Phi _{ef}^{\ast }(p)\)  has a non-trivial factor in
common with \(k+1\) . As we will see, the theorem is a tool that, together with [1]} and [2]}, allows us to control the structure of \(X_{x}\) .
